A right circular cylinder has a volume of $(6 + 2\sqrt{3})\pi\text{ cm}^3$ and a base radius of $(1 + \sqrt{3})\text{ cm}$. Find, without using a calculator, the height of the cylinder, in cm, in the form $(a + b\sqrt{3})$, where $a$ and $b$ are integers.

1. Set up the equation for the volume of the cylinder

Let the height of the cylinder be $h$.
Volume $V = \pi r^2 h$
$(6 + 2\sqrt{3})\pi = \pi (1 + \sqrt{3})^2 h$

2. Expand the square of the radius and simplify the equation

$(6 + 2\sqrt{3}) = (1 + 2\sqrt{3} + 3) h$
$6 + 2\sqrt{3} = (4 + 2\sqrt{3}) h$
$h = \frac{6 + 2\sqrt{3}}{4 + 2\sqrt{3}}$
$h = \frac{3 + \sqrt{3}}{2 + \sqrt{3}}$

3. Rationalise the denominator to find the height in the required form

$h = \frac{3 + \sqrt{3}}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}}$
$h = \frac{6 - 3\sqrt{3} + 2\sqrt{3} - 3}{4 - 3}$
$h = \frac{3 - \sqrt{3}}{1}$
$h = 3 - \sqrt{3}$